Question: $f(x) = \sqrt{ 9 - \lvert x \rvert }$ What is the domain of the real-valued function $f(x)$ ?
Explanation: $f(x)$ is undefined when the radicand (the expression under the radical) is less than zero. So we know that $9 - \lvert x \rvert \geq 0$ So $\lvert x \rvert \leq 9$ This means $x \leq 9$ and $x \geq -9$ ; or, equivalently, $-9 \leq x \leq 9$ Expressing this mathematically, the domain is $\{ \, x \in \RR \mid -9\leq x \leq9\, \}$.